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hw 6 solution

# hw 6 solution - Economics 314 Suggested Solutions to HW 6...

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Economics 314: Suggested Solutions to HW 6 TA: Yan Li March 14, 2009 Attention!!! During the lecture, I made a mistake about the calculation of C ot +1 . The formual for C ot +1 should be: c ot +1 = ( w t z yt )(1 + r t +1 ) z ot +1 where w t = (1 ) A t k t and r t +1 = t +1 k 1 t +1 Please refer to speci&c examples in Question 3,4,5. In partic- ular, we should use r t +1 instead of r t in calculating C ot +1 : Sorry for the confusion. 1 Question 1: 1. Given: F ( A t ; K t ; L t ) = A t K t L 1 t = ) y t = A t k t = 0 : 5 ; ± = 0 : 3 ; A t = 10 ; k 0 = 1 u ( c yt ; c ot +1 ) = ± ln c yt + (1 ± ) ln c ot +1 k t for t = 1 ; 2 : Transitional Equation: k t +1 = (1 ± )(1 ) A t k t 1

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k t +1 = 0 : 7 & 0 : 5 10 k 0 : 5 t = ) k 1 = 0 : 7 & 0 : 5 10 1 0 : 5 = ) k 1 = 3 : 5 k 2 = 0 : 7 & 0 : 5 10 k 0 : 5 1 = ) k 2 = 3 : 5 (3 : 5) 0 : 5 = ) k 2 = 6 : 5479 Growth Rates: g 0 ; 1 = k 1 ± k 0 k 0 100 = ) g 0 ; 1 = 3 : 5 ± 1 : 0 1 : 0 100 = 250% g 1 ; 2 = k 2 ± k 1 k 1 100 = ) g 1 ; 2 = 6 : 5479 ± 3 : 5 3 : 5 100 = 87 : 08% 2. In steady state, k t +1 = k t = k , so the Transitional Equation becomes: k = (1 ± )(1 ± ± ) A t k k = 0 : 7 & 0 : 5 10 k 0 : 5 k = 3 : 5 k & 0 : 5 k & 0 : 5 = 3 : 5 k = 12 : 25 y = Ak = 10 12 : 25 0 : 5 = 35 So the steady state k = 12 : 25 , and the steady state y = 35 . 2
Question 2: 1. = 0 : 5 ; ± = 0 : 7 ; A t = 10 (1 : 01) t ; k 0 = 1 The Transitional Equation is: k t +1 = (1 ± ± )(1 ± ) A t k t = 0 : 3 & 0 : 5 A t k t = 0 : 15 A t k t Economy is in steady state at t = 0 , so we must have k 1 k 0 = (1 : 01) 1 1 = (1 : 01) 2 = 1 : 02 = ) k 1 = 1 : 02 k 0 Plugging this into the transitional equation, we get: 1 : 02 k 0 = 0 : 15 10 k 0 : 5 0 Solving the above equation, we obtain: k 0 = 2 : 16 = ) k 1 = 1 : 02 k 0 = 1 : 02 2 : 16 = 2 : 206 y 0 = A 0 k & 0 = 10 2 : 16 0 : 5 = 14 : 70 The national saving rate at date 0 = N ( k 1 k 0 ) Ny 0 = 2 : 206 2 : 16 14 : 70 = 0 : 3% 2. If ± = 0 : 3 , The Transitional Equation is: k t +1 = (1 ± ± )(1 ± ) A t k t = 0 : 7 & 0 : 5 A t k t = 0 : 35 A t k t 3

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Economy is in steady state at t = 0 , so we must have k 1 k 0 = (1 : 01) 1 1 = (1 : 01)
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